In the United States, the United States Department of the Navy's General Order No. 135 issued in 1911 as a formal guide to naval terminology described a naval station as "any establishment for building, manufacturing, docking, repair, supply, or training under control of the Navy. It may also include several establishments". A naval base, by contrast, was "a point from which naval operations may be conducted".
From this, an educated guess would be that Cape Canaveral Space Force Station is auxiliary to headquarters at Patrick Space Force Base. Indeed, this is the case, but it is a bit of a messy history, to say the least.
Patrick SFB is the old Banana River Naval Air Station. Cape Canaveral SFS originates from the launch facilities used in the missile testing range that was developed in the Cape Canaveral geographic area and was an auxiliary base to when it got its first military designation together with Patrick (back then: Patrick Air Force Base).
Both were under US Air Force management and are now under US Space Force management.
The primary Cape Canaveral Space Force Station is to facilitate missile testing at the Eastern Range missile test range, while Patrick is more of a "general purpose" base and serves as headquarters. As such, CCSFS can still be considered auxiliary to Patrick SFB.
In the beginning, there was the Banana River Naval Air Station. There was nothing at the Cape area.
From "Moonport, chapter 1-3":
Cape Canaveral, better known as "the Cape," had been earmarked as a missile testing range in 1947.* [...] Cape Canaveral was a scenic but comparatively unsettled place.
With the footnote reading:
The selection was made by a Joint Chiefs of Staff committee. When the armed services went into rocketry in 1945, the Army stationed its launch team of German V-2 experts at White Sands, New Mexico - near the scene of Robert Hutchings Goddard's pioneering work in the 1930s. The southwestern desert proved too small for rockets, On 29 May 1947, a modified V-2 went the wrong way and landed in a cemetery south of Juarez, Mexico - one of the factors that decided the Joint Chiefs to move rocket experiments to the east coast of Florida.
The history of the Cape becoming a rocket testing site can be found in "A history of Cape Canaveral" by Cliff Lethbridge. It is important to remember that Von Braun's team was an US Army team, while the US Air Force was also doing rocket development. The Army's and Air Force's ideas did not always align, with various conflicts of interest as a result. Also recall that next to the US Air Force branch, the US Army and Navy branches maintain their own air forces, leading to confusing things as "Army air forces" and "Navy air forces", which are not to be conflated.
In 1947 the US Army became responsible for the development of a missile test range. Shortly after (2 weeks!) the US Air Force branch was established and inherited the missile group from the US Army branch, but also the Banana River Naval Air Station.
Again later that year, it was decided that all branches (Army, Navy and Air Force) were supposed to benefit from the missile range, so it was to be a joint program.
There were multiple options for replacing White Sands as testing ground, including sites in California (rejected by the Mexican president) and northern Washington (rejected due to its remoteness). The Cape (with the Banana River station as headquarters) was thus chosen as site in 1948, helped by the British offering to lease land for tracking stations in the Bahama's.
In 1949, the Joint Long Range Proving Ground Base (read: buildings and stuff) at the former Banana River Naval Air Station and the Joint Long Range Proving Ground (read: the actual missile range) became active under the management of an Air Force division, which dropped the "Joint" part already in early 1950: the base is now called Long Range Proving Ground Base and the missile range is now called Long Range Proving Ground.
On August 1, 1950, the base was renamed to Patrick Air Force Base (Patrick Space Force Base since 2020). The division responsible for the proving ground was renamed to Air Force Missile Test Center and the missile range was renamed to Florida Missile Test Range and later in 1958 to Atlantic Missile Range, to Eastern Test Range in 1964 and finally Eastern Range in 1992.
Note that the missile range covers a large area, starting from the launch area extending many thousands of miles, as shown in this contemporary map:
The launch area itself did not have a name until 1951, when it was designated Cape Canaveral Auxiliary Air Force Base. In 1955 this was renamed to Cape Canaveral Missile Test Annex.
By the end of the 1950s Cape Canaveral started to get crowded and when NASA's Saturn program started to take shape, they really needed more (bigger) launch complexes than the Cape could host. Many options were considered, but in the end Merrit Island was chosen. But:
Although Merritt Island, located to the north of Cape Canaveral, remained the prime site selection for NASA, a conflict with the Air Force developed. The Air Force wanted to reserve that land to allow expansion of Air Force rocket programs. [...] Conflicts with the Air Force continued over use of the range, because NASA sought their own jurisdiction over their own launch activities. The Air Force sought to continue control of the range at Cape Canaveral, with NASA as a tenant. The Air Force viewed Merritt Island as an extension of the range at Cape Canaveral, but NASA wanted their own property rights.
(from History of Cape Canaveral)
NASA got their way, but had to leave (a lot of never used) space for future Air Force developments.
In 1963, the NASA Launch Operations Center, comprising the launch facilities on Merrit Island and Cape Canaveral are renamed to John F. Kennedy Space Center by presidential executive order from president Lyndon B. Johnson, in honor of John F. Kennedy. The Air Force renames the Cape Canaveral Missile Test Annex to Cape Kennedy Air Force Station for the same reasons, taking effect in 1964.
Johson also renamed Cape Canaveral (the geographic area) to Cape Kennedy, but that didn't go over well with the local residents. After much arguing, this change was reverted in 1973. The NASA space center kept its name, but the Air Force station on the Cape was renamed to Cape Canaveral Air Force Station. At some point, the "Force" part was dropped and reinstated, until it got renamed to Cape Canaveral Space Force Station.
A full list of name changes can be found here.